Network Graphs and Tellegen’s Theorem

Network Graphs and Tellegen’s Theorem • The concepts of a graph • Cut sets and Kirchhoff’s current laws • Loops and Kirchhoff’s voltage laws • Tellegen’s Theorem

The concepts of a graph The analysis of a complex circuit can be perform systematically Using graph theories. Graph consists of nodes and branches connected to form a circuit. Fig. 1

The concepts of a graph Special graphs Fig. 2

The concepts of a graph G1 is a subgraph of Gif every node of G1 is the node of G and every branch of G1 is the branch of G Subgraph Fig. 3

The concepts of a graph Associated reference directions The kth branch voltage and kth branch current is assigned as reference directions as shown in fig. 4 Fig. 4 Graphs with assigned reference direction to all branches are called oriented graphs.

The concepts of a graph Fig. 5 Oriented graph Branch 4 is incident with node 2 and node 3 Branch 4 leaves node 3 and enter node 2

The concepts of a graph Incident matrix The node-to-branch incident matrix Aa is a rectangular matrix of nt rows and b columns whose element aik defined by If branch k leaves node i If branch k enters node i If branch k is not incident with node i

The concepts of a graph For the graph of Fig.5 the incident matrix Aa is

Cutset and Kirchhoff’s current law If a connected graph were to partition the nodes into two set by a closed gussian surface , those branches are cut set and KCL applied to the cutset Fig. 6 Cutset

Cutset and Kirchhoff’s current law A cutset is a set of branches that the removal of these branches causes two separated parts but any one of these branches makes the graph connected. An unconnected graph must have at least two separate part. Fig. 7

Cutset and Kirchhoff’s current law Fig. 8

Cutset and Kirchhoff’s current law Fig. 9

(c) Fig. 9

Cutset and Kirchhoff’s current law • For any lumped network , for any of its cut sets, and at any time, the algebraic sum of all branch currents traversing the cut-set branches is zero. From Fig. 9 (a) for all And from Fig. 9 (b) for all

Cutset and Kirchhoff’s current law Cut sets should be selected such that they are linearly independent. III Fig. 10 Cut sets I,II and III are linearly dependent

Cutset and Kirchhoff’s current law Cut set I Cut set II Cut set III KCLcut set III= KCLcut set I + KCLcut set II

Loops and Kirchhoff’s voltage laws A Loop L is a subgraph having closed path that posses the following properties: • The subgraph is connected • Precisely two branches of L are incident with each node Fig. 11

Loops and Kirchhoff’s voltage laws Fig. 12 Cases I,II,III and IV violate the loop Case V is a loop

Loops and Kirchhoff’s voltage laws • For any lumped network , for any of its loop, and at any time, the algebraic sum of all branch voltages around the loop is zero. Example 1 Write the KVL for the loop shown in Fig 13 KVL for all Fig. 13

Tellegen’s Theorem • Tellegen’s Theorem is a general network theorem • It is valid for any lump network For a lumped network whose element assigned by associate reference direction for branch voltage and branch current The product is the power delivered at time by the network to the element If all branch voltages and branch currents satisfy KVL and KCL then = number of branch

Tellegen’s Theorem Suppose that and is another sets of branch voltages and branch currents and if and satisfy KVL and KCL Then and

Tellegen’s Theorem Applications Tellegen’s Theorem implies the law of energy conservation. Since “The sum of power delivered by the independent sources to the network is equal to the sum of the power absorbed by all branches of the network”.

Applications • Conservation of energy • Conservation of complex power • The real part and phase of driving point impedance • Driving point impedance

Conservation of Energy For all t “The sum of power delivered by the independent sources to the network is equal to the sum of the power absorbed by all branches of the network”.

Conservation of Energy • Resistor • Capacitor • Inductor For kth resistor For kth capacitor For kth inductor

Conservation of Complex Power = Branch Voltage Phasor = Branch Current Phasor = Branch Current Phasor Conjugate

Conservation of Complex Power

The real part and phase of driving point impedance

From Tellegen’s theorem, and let P = complex power delivered to the one-port by the source

Taking the real part All impedances are calculated at the same angular frequency i.e. the source angular frequency

Driving Point Impedance C L R

Exhibiting the real and imaginary part of P Average power dissipated Average Magnetic Energy Stored Average Electric Energy Stored

From

Driving Point Impedance Given a linear time-invariant RLC network driven by a sinusoidal current source of 1 A peak amplitude and given that the network is in SS, The driven point impedance seen by the source has a real part = twice the average power Pavand an imaginary part that is 4w times the difference of EM and EE

Network Graphs and Tellegen’s Theorem